Sure, let's work through the problem step-by-step. We need to fit a quadratic model \( y = b_0 + b_1 x + b_2 x^2 \) to the given data:
[tex]\[
\begin{array}{c|c|c|c|c|c}
\hline
x & 1 & 2 & 3 & 4 & 5 \\
\hline
y & 5 & 20 & 45 & 75 & 110 \\
\hline
\end{array}
\][/tex]
Given the data transformation \( v = x - 3 \) and \( w = \frac{y - 45}{5} \), let's first compute the transformed values:
[tex]\[
\begin{array}{c|c|c|c|c|c}
\hline
x & 1 & 2 & 3 & 4 & 5 \\
\hline
v & -2 & -1 & 0 & 1 & 2 \\
\hline
\end{array}
\][/tex]
[tex]\[
\begin{array}{c|c|c|c|c|c}
\hline
y & 5 & 20 & 45 & 75 & 110 \\
\hline
w & -8 & -5 & 0 & 6 & 13 \\
\hline
\end{array}
\][/tex]
Next, we compute the necessary sums for the transformed values \(v\) and \(w\). We need the following sums:
[tex]\[
\sum v, \quad \sum v^2, \quad \sum v^3, \quad \sum v^4, \quad \sum w, \quad \sum vw, \quad \sum v^2 w
\][/tex]
From the problem statement and provided results:
1. \( n = 5 \) (number of data points)
2. \( \sum v = 0 \)
3. \( \sum v^2 = 10 \)
4. \( \sum v^3 = 0 \)
5. \( \sum v^4 = 34 \)
6. \( \sum w = 6.0 \)
7. \( \sum vw = 53.0 \)
8. \( \sum v^2 w = 21.0 \)
These values are confirmed as follows:
\begin{align}
\sum v & = -2 + (-1) + 0 + 1 + 2 = 0 \\
\sum v^2 & = (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 = 4 + 1 + 0 + 1 + 4 = 10 \\
\sum v^3 & = (-2)^3 + (-1)^3 + 0^3 + 1^3 + 2^3 = -8 - 1 + 0 + 1 + 8 = 0 \\
\sum v^4 & = (-2)^4 + (-1)^4 + 0^4 + 1^4 + 2^4 = 16 + 1 + 0 + 1 + 16 = 34 \\
\sum w & = -8 + (-5) + 0 + 6 + 13 = 6 \\
\sum vw & = (-2)(-8) + (-1)(-5) + (0)(0) + (1)(6) + (2)(13) = 16 + 5 + 0 + 6 + 26 = 53 \\
\sum v^2 w & = (-2)^2(-8) + (-1)^2(-5) + (0)^2(0) + (1)^2(6) + (2)^2(13) = 4(-8) + 1(-5) + 0 + 1(6) + 4(13) \\
& = -32 - 5 + 0 + 6 + 52 = 21
\end{align}
Using these sums, we can form the system of equations as follows:
[tex]\[
\begin{pmatrix}
n & \sum v & \sum v^2 \\
\sum v & \sum v^2 & \sum v^3 \\
\sum v^2 & \sum v^3 & \sum v^4
\end{pmatrix}
\begin{pmatrix}
b_0 \\
b_1 \\
b_2
\end{pmatrix}
=
\begin{pmatrix}
\sum w \\
\sum vw \\
\sum v^2 w
\end{pmatrix}
\][/tex]
Substituting the computed values gives:
[tex]\[
\begin{pmatrix}
5 & 0 & 10 \\
0 & 10 & 0 \\
10 & 0 & 34
\end{pmatrix}
\begin{pmatrix}
b_0 \\
b_1 \\
b_2
\end{pmatrix}
=
\begin{pmatrix}
6 \\
53 \\
21
\end{pmatrix}
\][/tex]
So the system of equations is:
[tex]\[
\begin{aligned}
5b_0 + 0b_1 + 10b_2 &= 6 \\
0b_0 + 10b_1 + 0b_2 &= 53 \\
10b_0 + 0b_1 + 34b_2 &= 21
\end{aligned}
\][/tex]
This is the system of equations that yields the least squares regression coefficients for the quadratic model after applying the given data transformations.
